A combination is a way of selecting several elements from a larger set of elements, where the order of the elements is not important.

Combinations can be with repetition and without repetition.

Combinations with repetition

Combinations with repetition are calculated according to the formula

Formula for calculating combinations with repetition

where is

K - combination symbol,
n - total number of elements, and
r - the number of elements to be selected.

Combinations without repetition

Combinations without repetition represent the number of ways in which it is possible to select r elements from a set containing n elements, where the order of the elements is not important, and the selected elements do not participate in the new selection. They are calculated using the formula

Formula for calculating combinations without repetition

where is

K - combination symbol,
n - total number of elements, and
r - number of elements to be chosen.

This expression is also called the binomial coefficient.

Example: Lotto

In the game Lotto 7 out of 39 7 numbers are drawn out of a possible 39. The order of the numbers drawn does not affect the prize.

How many possible combinations are there?

Answer: There are 39!/(7!*32!) = 15,380,937 possible number combinations.